Storage in Barn for winter.

If on dirt or grass, yes put lumber under the tires for the sole purpose of keeping the tires from sinking. On Asphalt and concrete, nothing is needed, unless you just want to spread the weight of the camper over a larger foot print. Gravel is 50/50. It depends upon how solid the gravel bed is. If it's not very deep, and tires are subject to sinking with mud under the gravel, the lumber under the tires is probably a good thing. If it's packed solid and deep, lumber is not really needed. Previous house, previous TT, parked on gravel, always on treated lumber. Current house, one side on lumber, the other not. (the driveway has a 4 inch slope side to side for deliberate drainage one direction). So 5er needs lumber under one side to make it more level.

Plugged in? Ours? 24x7x365. It depends on your converter.

Something to think about? If the pole barn is an open barn, consider keeping the camper in ready mode, so if you want to spend week-end in it, you can. Power up the heat and enjoy watching television and sleeping in it. Don't need water. Take a gallon jug of water so if you need some, put a port-a-potty in the bathroom and dump in the house toilet every day, rinse and use it again, snuggle in bed with .... well... your choice ;) take some microwave pop corn and enjoy some time in it. Under a roof, the camper should do very, very nice in cold weather. Enjoy. Do dishes in the house, take trash out at the end of your weekend, and enjoy the camper throughout the winter this way.